Risk-Neutral Exercise Probabilities for Caplets and Floorlets

Summary

The document asks whether the Black model’s N(d2) represents the risk-neutral probability that a vanilla caplet finishes in the money, and what probability applies to a floorlet. The response interprets the question as one about the probability of a digital option expiring in the money. Under the stated Black-model setup, it affirms N(d2) for the caplet and N(−d2) for the floorlet, following the call and put relationship.

The answer is brief and offers no derivation or numerical example, so it provides a result rather than a full explanation of the model’s probability measure and assumptions. It also flags a limitation of the standard Black rate model: its lognormal assumption excludes negative rates. That caveat matters when applying the exercise-probability interpretation to interest-rate markets where negative rates may occur; an alternative model or convention may be needed in such settings.

Key ideas

• In the Black framework, N(d2) is given as the risk-neutral probability of caplet exercise.
• The corresponding floorlet probability is N(−d2), analogous to a digital put.
• The response frames exercise probability as the chance that the relevant rate finishes in the payoff region.
• The standard lognormal Black rate assumption does not accommodate negative rates.

Tags

# Exercise Probabilities Vanilla Cap/Floor


# Exercise Probabilities Vanilla Cap/Floor












When looking at the discounted pay-off formulas of a vanilla caplet and a vanilla floorlet

$\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_k-r_{cap},0)$

$\frac{\Delta\tau}{1+r_k\Delta\tau}\max(r_{floor}-r_k,0)$

$r_{cap/floor} = $ cap/floor rate between $t_k$ and $t_{k+1}$

$r_{k} = $ realised interest rate between $t_k$ and $t_{k+1}$

$\tau = $ reset period

then my intuition tells me that $N(d_2)$ could possibly be the Black risk-neutral exercise probability of the caplet.

- Is this assumption correct?

- If it is, what would be the correct risk-neutral probability of the floorlet? My guess is that it wouldn't be $N(-d_2)$, but I'm not sure.

Thanks in advance.

## Answer by FinanceGuyThatCantCode (score 1)

https://quant.stackexchange.com/a/34076

Seems like you are asking for the risk neutral probability that a digital option on the respective rates will be ITM at expiration. Also, since you say you are assuming Black's model, you are implicitly assuming that rates can never go below zero - which we know to be wrong of course. As such, your intuition that the probability of the caplet exercising is indeed $N(d_2)$. Also, the probability of the floorlet exercising is $N(-d_2)$ just like any other digital put. Unless I am missing something?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.